EXERCISE - 01 CHECK YOUR GRASP

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1 J-Mathematics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). The maximum value of the sum of the A.P. 0, 8, 6,,... is Let T r be the r th term of a A.P. for r =,,,... If for some positive itegers m, we have T m & T, m the T equals - m m 0 m. The iterior agles of a covex polygo are i AP. The smallest agle is 0 & the commo differece is. Fid the umber of sides of the polygo oe of these. The first term of a ifiitely decreasig G.P. is uity ad its sum is S. The sum of the squares of the terms of the progressio is - S S S S S S S. A particle begis at the origi ad moves successively i the followig maer as show, uit to the right, / uit up, / uit to the right, /8 uit dow, /6 uit to the right etc. The legth of each move is half the legth of the previous move ad movemet cotiues i the zigzag maer idefiitely. The co-ordiates of the poit to which the zigzag coverges is - (/, /) (/, /) (/, /) (, /) y 0 / / /8 /6 x NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6 6. Let a be the th term of a G.P. of positive umbers. Let commo ratio of the G.P. is - 00 a = & 00 a = such that. The the 7. If p, q, r i harmoic progressio ad p & r be differet havig same sig the the roots of the equatio px + qx + r = 0 are - real ad equal real ad distict irratioal imagiary 8. If x >, y >, z > are i G.P., the x, y, z are i - A.P. H.P. G.P. oe of above 9. If l (a + c), l (c a), l (a b + c) are i A.P., the : a, b, c are i A.P. a, b, c are i G.P. a, b, c are i A.P a, b, c are i H.P. 7

2 J-Mathematics 0. If the (m + ) th, ( +) th & (r + ) th terms of a AP are i GP & m,, r are i HP, the the ratio of the commo differece to the first term of the AP is - oe of these. The sum of roots of the equatio ax + bx + c = 0 is equal to the sum of squares of their reciprocals. The bc, ca ad ab are i - AP GP HP oe of these. The quadratic equatio whose roots are the A.M. ad H.M. betwee the roots of the equatio, x x + = 0 is - x x + 0 = 0 x 9x + 0 = 0 x x + = 0 x + x + = 0. If the sum of the first atural umbers is / times the sum of the their squares, the the value of is Suppose p is the first of ( > ) AM's betwee two positive umbers a ad b, the value of p is - a b a b b a b a. If 0 a a b c c b ad a, b, c are ot i A.P., the - a, b, c are i G.P. a, b, c are i A.P. b a,,c are i H.P. a, b, c are i H.P The sum to terms of the series... is If... + to 8. If 96 s s r r a b c, the... + to is equals to - 90, the fid the value of a + b + c If a, b, c are positive umbers i G.P. ad sides of a triagle which is - c b log, log a c ad a log b oe of these are i A.P., the a, b, c forms the equilateral right agled isosceles oe of these SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS) 0. If sum of terms of a sequece is give by S = + 7 & t r represets its r th term, the - t 7 = t = 7 t 0 = t 8 = 0 NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

3 J-Mathematics. If 0 harmoic meas H, H, H... H 0 are iserted betwee 7 ad, the - H = 7 H = 7 H = 7 H 0 = 7 9. If t be the th term of the series , the - t + = t 7 = 7 + t 0 = 0 t 00 = 0 +. Idicate the correct alterative(s), for 0, if the - x cos, y si ad 0 0 z cos si, 0 xyz = xz + y xyz = xy + z xyz = x + y + z xyz = yz + x NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6 ANSWR KY BRAIN TASRS X RCIS - Q u e A s. C C A B B A D B D C Q u e A s. A B C A D B A A D A, D Q u e. A s. A, D A, C B, C 9

4 J-Mathematics XRCIS - 0 BRAIN TASRS SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS). Cosider a A.P. with first term a ad the commo differece d. Let S k deote the sum of the first K terms. Let S S kx x is idepedet of x, the - a = d/ a = d a = d oe of these. Let,, be the roots of the equatio x + ax + bx + c = 0. If,, are i H.P. the is equal to - c/b c/b a a 9. ( r ) is equal to - r F H G I K J r sum of first ie atural umbers sum of first te atural umbers. For the A.P. give by a, a,..., a,..., with o-zero commo differece, the equatios satisfied are- a + a + a = 0 a a + a = 0 a + a a a = 0 a a + 6a a + a = 0. If a, a, a,...,a 0, b are i A.P. ad a, g, g,...g 0, b are i G.P. ad h is the H.M. betwee a ad b, the a a... a0 a a... a9 a a6... g g g g g g is h h 0 h h 6. The sum of the first terms of the series is ( ), whe is eve. Whe is odd, the sum is - ( ) ( )( ) 6 0 ( ) ( ) 7. If ( a) + ( b) = ( c), where each set of paretheses cotais the sum of cosecutive odd itegers as show such that - (i) a + b + c =, (ii) a > 6 If G = Max{a, b, c} ad L = Mi{a, b, c}, the - G L = b a = G L = 7 a b = 8. If a, b ad c are distict positive real umbers ad a + b + c =, the ab + bc + ca is - equal to less tha greater tha ay real umber 9. Let p, q, r R + ad 7 pqr (p + q + r) ad p + q + r = the p + q + r is equal to - 6 oe of these 0. The sum of the first 00 terms commo to the series 7,,,... ad 6,, 6,...is NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

5 . If a, b, c are positive such that ab c = 6 the least value of F I a HG b c K J is - 6 J-Mathematics. If a, a,...a R + ad a.a... a = the the least value of ( a a )( a a )...( a a ) is - data iadequate. Let a, a, a,... ad b, b, b,... be arithmetic progressio such that a =, b = 7 ad a 00 + b 00 = 00, the - The commo differece i progressio 'a i ' is equal but opposite i sig to the commo differece i progressio 'b j '. a + b = 00 for ay. (a + b ), (a + b ), (a + b ),... are i A.P. 00 r (a b ) 0 r r. If the AM of two positive umbers be three times their geometric mea the the ratio of the umbers is - 7. If first ad ( ) th terms of a A.P., G.P. ad H.P. are equal ad their th terms are a, b, c respectively, the - a + c = b a b c a + c = b b = ac 6. Let a, x, b be i A.P. ; a, y, b be i G.P. ad a, z, b be i H.P. If x = y + ad a = z the - y = xz x > y > z a = 9, b = a = 9, b = 7. The p th term T p of H.P. is q(q + p) ad q th term T q is p(p + q) whe p >, q >, the - NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6 T p + q = pq T pq = p + q T p + q > T pq T pq > T p+q 8. a, b, c are three distict real umbers, which are i G.P. ad a + b + c = xb, the - x < < x < < x < x > 9. Let a, a,..., a 0 be i A.P. & h, h,...h 0 be i H.P.. If a = h = & a 0 = h 0 = the a h 7 is - 6 BRAIN TASRS ANSWR KY X RCIS - Q u e A s. A A A, C B, D C A A, D B C A Q u e A s. C A A,B,C,D C, D B, D A,B,C A,B,C A, D D

6 J-Mathematics XRCIS - 0 MISCLLANOUS TYP QUSTIONS FILL IN TH BLANKS. The sum of terms of two A.P. s are i the ratio of ( + 7) : ( + ). The ratio of their 9th term is.. The sum of the first ietee terms of a A.P. a, a, a... if it is kow that a + a 8 + a + a 6 =, is.. If x R ad the umbers ( +x + x ), a/, ( x + x ) form a A.P. the a must lie i the iterval.. If upto =, the =.. Whe 9 th term of a A.P. is divided by its d term the quotiet is & whe th term is divided by the 6 th term, the quotiet is ad remaider is. The first term ad the commo differece of the A.P. are & respectively. 6. The sum to ifiity of the series is equal to. 7. If si (x y), si x ad si (x + y) are i H.P., the si x. sec y =. MATCH TH COLUMN Followig questios cotais statemets give i two colums, which have to be matched. The statemets i Colum-I are labelled as A, B, C ad D while the statemets i Colum-II are labelled as p, q, r ad s. Ay give statemet i Colum-I ca have correct matchig with ON statemet i Colum-II.. Colum-I Colum-II If a i 's are i A.P. ad a + a + a + a + a 7 = 0, a (p) is equal to Sum of a ifiite G.P. is 6 ad it's first term is. (q) the harmoic mea of first ad third terms of G.P. is If roots of the equatio x ax + bx + 7= 0, are i G.P. (r) with commo ratio, the a + b is equal to If the roots of x 8x + ax + bx + 6 = 0 are (s) 6/ positive real umbers the a is. Colum-I Colum-II th term of the series,,, 7, 6, 79,... (p) terms is equal to (q) + + sum to terms of the series, 7,,,... is (r) ( + ) coefficiet of x i x(x )(x )... (x ) is (s) ASSRTION & R ASON ( ) These questios cotais, Statemet-I (assertio) ad Statemet-II (reaso). Statemet-I is true, Statemet-II is true ; Statemet-II is correct explaatio for Statemet-I. Statemet-I is true, Statemet-II is true ; Statemet-II is NOT a correct explaatio for Statemet-I. Statemet-I is true, Statemet-II is false. Statemet-I is false, Statemet-II is true. NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

7 J-Mathematics NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6 a b. Statemet-I : If a, b, c are three distict positive umber i H.P., the a b + c b c b > B e c a u s e Statemet-II : Sum of ay umber ad it's reciprocal is always greater tha or equal to. A B C D. Statemet-I : If x y = 6(x, y > 0), the the least value of x + y is 0 B e c a u s e m m m m ma m a Statemet-II : If m, m N, a, a > 0 the (a a ) m m a = a. A B C D. Statemet-I : For N, > + ( ) B e c a u s e Statemet-II : G.M. > H.M. ad (AM) (HM) = (GM) A B C D. Statemet-I : If a, b, c are three positive umbers i G.P., the a b c. abc ab bc ca ad equality holds whe = abc B e c a u s e Statemet-II : (A.M.) (H.M.) = (G.M.) is true for ay set of positive umbers. A B C D. Statemet-I : th term (T ) of the sequece (, 6, 8, 0, 7, 6,...) is a + b + c + d, ad 6a + b d is =. B e c a u s e Statemet-II If the secod successive differeces (Differeces of the differeces) of a series are i A.P., the T is a cubic polyomial i. A B C D 6. Statemet-I : The format of th term (T ) of the sequece (,,, 0...) is a + b + c. B e c a u s e Statemet-II : If the secod successive differeces betwee the cosecutive terms of the give sequece are i G.P., the T = a + b + cr, where a, b, c are costats ad r is commo ratio of G.P. A B C D COMPRHNSION BASD QUSTIONS Comprehesio # There are + terms i a sequece of which first + are i Arithmetic Progressio ad last + are i Geometric Progressio the commo differece of Arithmetic Progressio is ad commo ratio of Geometric Progressio is /. The middle term of the Arithmetic Progressio is equal to middle term of Geometric Progressio. Let middle term of the sequece is T m ad T m is the sum of ifiite Geometric Progressio whose F I 9 sum of first two terms is HG K J ad ratio of these terms is 6. O t he basis of above i format io, a swer t he fol low i g que st ios :. Number of terms i the give sequece is equal to oe. Middle term of the give sequece, i.e. T m is equal to - 6/7 /7 8/7 6/9. First term of give sequece is equal to - 8/7, 0/7 6/7 6/7 8/7

8 J-Mathematics. Middle term of give A. P. is equal to - 6/7 0/7 78/7. Sum of the terms of give A. P. is equal to - 6/7 7 6 Comprehesio # : If a i > 0, i =,,,... ad m, m, m,..., m be positive ratioal umbers, the ma ma... m a m m... m m m m a a...a is called weighted mea theorem where A* = ma ma... m a m m... m m m m G* = a a...a /(m m... m ) /(m m...m ) = Weighted arithmetic mea (m m... m) m m m... a a a = Weighted geometric mea ad H* = i.e., A* G* H* m m... m m m m... a a a = Weighted harmoic mea Now, let a + b + c = (a, b, c > 0) ad x y = (x > 0, y > 0) O t he basis of above i format io, a swer t he fol low i g que st ios :. The greatest value of ab c is Which statemet is correct a b c a b c a b c a b c. The least value of x + y + is - greater tha less tha. Which statemet is correct - x y x y x y xy x y Fill i the Blaks x y x y xy x y xy x y MISCLLANOUS TYP QUSTION ANSWR KY X RC IS -. :. 06. [, ]. /8. a = d = ± Match the Colum. (q), (s), (p), (r). (q), (p), (p), (r) Assertio & Reaso. C. A. C. C. A 6. B Comprehesio Based Que st ios Comprehesio # :. C. C. B. A. D Comprehesio # :. C. C. B. B NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

9 J-Mathematics XRCIS - 0 [A] CONCPTUAL SUBJCTIV XRCIS. Give that a x = b y = c z = d u & a, b, c, d are i GP, show that x, y, z, u are i HP.. There are AM s betwee & such that 7th mea : ( )th mea = : 9, the fid the value of.. Fid the sum of the series, to terms.. If the p th, q th & r th terms of a AP are i GP. Show that the commo ratio of the GP is q r p q.. xpress the recurrig decimal as a ratioal umber usig cocept of ifiite geometric series. 6. If oe AM a & two GM s p & q be iserted betwee ay two give umbers the show that p + q = apq. 7. Fid three umbers a, b, c betwee & 8 which satisfy followig coditios : (i) their sum is (ii) the umbers, a, b are cosecutive terms of a AP & (iii) the umbers b, c, 8 are cosecutive terms of a GP. 8. Fid the sum of the first terms of the series : 9. Let a, a, a... a be a AP. Prove that :... = a a a a a a a a a a a a a a 0. The harmoic mea of two umbers is. The arithmetic mea A & the geometric mea G satisfy the relatio A + G² = 7. Fid the two umbers.. Prove that : (ab + xy)(ax + by) abxy where a, b, x, y R +. If a, b, c R + & a + b + c = ; the show that ( a)( b)( c) 8abc. If a, b, c are sides of a scalee triagle the show that (a + b + c) > 7 (a + b c)(b + c a)(c + a b) NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6. For positive umber a, b, c show that bc ac ab a b c a b c. The odd positive umbers are writte i the form of a triagle fid the sum of terms i th row. CONCPTUAL SUBJCTIV XRCIS ANSWR KY X RC IS - ( A).. S = (7/8)( ). / 7. a =, b = 8, c = ,.

10 J-Mathematics XRCIS - 0 [B] BRAIN STORMING SUBJCTIV XRCIS. I a A.P. & a H.P. have the same first term, the same last term & the same umber of terms; prove that the product of the r th term from the begiig i oe series & the r th term from the ed i the other is idepedet of r.. Sum the followig series to terms ad to ifiity : (a)... (b) Fid the value of the sum. Fid the sum i. i j k 6 r (r + ) (r + ) (r + ) r r s rs where rs is zero if r s & rs is oe if r = s. r s j (c) r r. If there be 'm' A.P s begiig with uity whose commo differece is,,... m. Show that the sum of their th terms is (m/) (m m + + ). 6. If a, a, a... a are i H.P., the prove that a a + a a a - a = ( ) a a. 7. If a, b, c are i H.P., b, c, d are i G.P. & c, d, e are i A.P., the Show that e = ab²/(a b)². 8. The value of x + y + z is, if a, x, y, z, b are i A.P. while the value of ; (/x)+(/y)+(/z) is / if a, x, y, z, b are i H.P. Fid a & b Prove that the sum of the ifiite series If a, b, c be i G.P. & log a, log c, log b be i A.P., the show that the commo differece of the A.P. must c b a be /.. Fid the sum to terms : (a) x x... x (x ) (x ) (x ) (x ) (x ) a a a... (b) a a a a a a. I a G.P., the ratio of the sum of the first eleve terms to the sum of the last eleve terms is /8 ad the ratio of the sum of all the terms without the first ie to the sum of all the terms without the last ie is. Fid the umber of terms i the G.P.. Prove that the umber is a per fect square of the umber digits (-) digits (-) digits. Fid the th term ad the sum to '' terms of the series : (a) (b) a b c. If a, b, c are three positive real umber the prove that : b c a c a b 6. If a, b, c are the sides of a triagle ad s = a b c, the prove that 8(s a)(s b)(s c) abc. BRAIN STORMING SUBJCTIV XRCIS ANSWR KY XRCI S - ( B ). (a). 6 (6 ). (a), 6( )( ) (b) ( )( )( )( ) (c),. [(+)(+)]/6 8. a =, b = 9 or b =, a = 9 x (x )(x )...(x ) (b) ( a )( a )...( a ). = 8. (a) + ; + (b) + + ; ( )( ) 6 NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

11 J-Mathematics XRCIS - 0 [A] J-[MAIN] : PRVIOUS YAR QUSTIONS x. If, log, log (. x ) are i A.P. the x equals. [AI 00] () log () log () log () log. Sum of ifiite umber of terms i G.P. is 0 ad sum of their square is 00. The commo ratio of G.P. is- [AI 00] () () / () 8/ () /. Fifth term of a G.P. is, the the product of its 9 terms is- [AI 00] () 6 () () 0 () Noe of these. The sum of the series = [AI 00] () 00 () () () 0. Let T r be the rth term of a A.P. whose first term is a ad commo differece is d. If for some positive itegers m,, m, T m = ad T = m, the a d equals [AI 00] () 0 () () m () m 6. If AM ad GM of two roots of a quadratic equatio are 9 ad respectively, the this quadratic equatio is- [AI 00] () x 8x + 6 = 0 () x + 8x 6 = 0 () x + 8x + 6 = 0 () x 8x 6 = 0 7. If a, a, a,... a,... are i G.P. the the value of the determiat log a log a log a log a log a log a log a log a log a 6 7 8, is- () 0 () () () [AI 0, 0] NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6 8. If x = a, y = 0 b, z = 0 C where a, b, c are i A.P. ad a <, b <, 0 c < the x, y, z are i- [AI 00] () HP () Arithmetic - Geometric Progressio () AP () GP 9. Let a, a, a,... be terms of a A.P. If () 7 () a a... ap p, p q the a a... a q q () a6 a equals-[ai-006] 0. If a, a,..., a are i H.P., the the expressio a a + a a a a is equal to-[ai-006] () 7 () a a () ( )a a () (a a ) () ( )(a a ). I a geometric progressio cosistig of positive terms, each term equals the sum of the ext two terms. The the commo ratio of this progressio equals- [AI- 007] () () () ( ) () ( ) 7

12 J-Mathematics. The first two terms of a geometric progressio add up to. The sum of the third ad the fourth terms is 8. If the terms of the geometric progressio are alterately positive ad egative, the the first term is [AI 008] () () () () 6 0. The sum to ifiity of the series... is :- [ AI ] () () 6 () (). A perso is to cout 00 currecy otes. Let a deote the umber of otes he couts i the th miute. If a = a =... = a 0 = 0 ad a 0, a,... are i a AP with commo differece, the the time take by him to cout all otes is :- [AI-00] () miutes () miutes () miutes () miutes. A ma saves Rs. 00 i each of the first three moths of his service. I each of the subsequet moths his savig icreases by Rs. 0 more tha the savig of immediately previous moth. His total savig from the start of service will be Rs. 00 after :- [ AI- 0 ] () 0 moths () moths () 8 moths () 9 moths Let a be the th term of a A.P. If a r ad a r, the the commo differece of the A.P. is: r r [ AI- 0 ] () 00 () () 00 () 7. Statemet : The sum of the series + ( + + ) + ( ) + ( ) ( ) is k (k ), for ay atural umber. [ AI- 0 ] k Statemet : () Statemet is true, Statemet is false. () Statemet is false, Statemet is true. () Statemet is true, Statemet is true ; Statemet is a correct explaatio for Statemet. () Statemet is true, Statemet is true ; Statemet is ot a correct explaatio for Statemet. 8. If 00 times the 00 th term of a A.P. with o-zero commo differece equals the 0 times its 0 th term, the the 0 th term of this A.P. is : [AI-0] () zero () 0 () 0 times its 0 th term () 0 9. The sum of first 0 terms of the sequece 0.7, 0.77, 0.777,..., is : [J-MAIN 0] () 7 ( ) 8 PRVIOUS YARS QUSTIONS () 7 ( ) 9 8 () 7 ( ) 8 ANSWR KY () 7 ( ) XRCIS- [A] NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

13 J-Mathematics XRCIS - 0 [B] J-[ADVANCD] : PRVIOUS YAR QUSTIONS. ( a ) Cosider a i fi ite geometric serie s w ith first term a ad commo rat io r. If the sum is ad the secod term is /, the - [J 000, Screeig, +M out of ] 7 a, r a =, 7 r 8 a, r a =, r ( b ) If a, b, c, d are positive real umbers such that a + b + c + d =, the M = (a + b) (c + d) satisfies the relatio - 0 M M M M ( c ) The fourth power of the commo differece of a arithmetic progressio with iteger etries is added to the product of ay four cosecutive terms of it. Prove that the resultig sum is the square of a iteger. [J 000, Mais, M out of 00]. ( a ) Let, be the roots of x - x + p = 0 ad, be the roots of x - x + q = 0. If,,, are i G.P.,., the the iteger values of p ad q respectively, are - [J 00 Screeig ++M out of ],, 6, 6, ( b ) If the sum of the first terms of the A.P.,, 8... is equal to the sum of the first terms of the A.P. 7, 9, 6,... the equals - 0 ( c ) Let the positive umbers a, b, c, d be i A.P. The abc, abd, acd, bcd are ot i A.P./G.P./H.P. i G.P. i H.P. i A.P. ( d ) Let a, a... be positive real umbers i G.P.. For each, let A, G, H, be respectively, the arithmetic mea, geometric mea ad harmoic mea of a, a, a,... a. Fid a expressio for the G.M. of G, G,...G i terms of A, A,...A, H, H,...H [J 00 (Mais) ; M] NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6. ( a ) Suppose a, b, c are i A.P. ad a, b, c are i G.P. If a < b < c ad a b c, the the value of a is - [J 00 ( Screei g), M] ( b ) Let a, b be positive real umbers. If a, A, A, b are i A.P. ; a, G, G, b are i G.P. ad a, H, H, b are i H.P., show that GG A A (a b)(a b). [J 00, Mais, M out of 60] H H H H 9ab. If a, b, c are i A.P., a, b, c are i H.P., the prove that either a = b = c or a, b, c form a G.P. [J 00, Mais, M out of 60]. If a, b, c are positive real umbers, the prove that [( + a)( + b)( + c)] 7 > 7 7 a b c. [J 00, M] 6. The first term of a ifiite geometric progressio is x ad its sum is. The - [J 00] 0 x 0 0 x 0 0 < x < 0 x > 0 9

14 J-Mathematics F HG I K J 7. If total umber of rus scored i matches is ( + ) where >, ad the rus scored i the k th match are give by k. + k, where k. Fid. [J-0, Mais-M out of 60] 8. I quadratic equatio ax + bx + c = 0, if are roots of equatio, = b ac ad +, +, + are i G.P. the [J 00 (screei g)] 0 b= 0 c= 0 = 0 F H G I K J F H G I K J F H G I K J 9. If a...( ) ad b = a the fid the miimum atural umber 0 such that b > a 0 [J 006, 6M out of 8] Comprehesio Based Que st io Comprehesio # Let V r deote the sum of first r terms of a arithmetic progressio (A.P.) whose first term is r ad the commo differece is (r ). Let T r = V r + V r ad Q r = T r + T r for r =,, The sum V + V V is : [J 007, M] ( + ) ( + ) ( + ) ( + + ) ( + ) ( + ). T r is always : [J 007, M] a odd umber a eve umber a prime umber a composite umber. Which oe of the followig is a correct statemet? [J 007, M] Q,Q,Q,...are i A.P. with commo differece Q,Q,Q,...are i A.P. with commo differece 6 Q,Q,Q,...are i A.P. with commo differece Q = Q = Q =... Comprehesio # Let A, G, H deote the arithmetic, geometric ad harmoic meas, respectively, of two distict positive umbers. For, let A ad H has arithmetic, geometric ad harmoic meas as A, G, H respectively :. Which oe of the followig statemets is correct? [J 007, M] G > G > G >... G < G < G <... G = G = G =... G < G < G <... ad G > G > G 6 >.... Which oe of the followig statemets is correct? [J 007, M] A > A > A >... A < A < A <... A > A > A >... ad A < A < A 6 <... A < A < A <... ad A > A > A 6 >.... Which oe of the followig statemets is correct? [J 007, M] H > H > H >... H < H < H <... H > H > H >... ad H < H < H 6 >... H < H < H <... ad H > H > H 6 >... 0 NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6

15 J-Mathematics 6. Suppose four distict positive umbers a, a, a, a are i G.P. Let b = a, b = b + a, b = b + a ad b = b + a. Statemet -I : The umbers b, b, b, b are either i A.P. or i G.P. a d Statemet -II : The umbers b, b, b, b are i H.P. [J 008, M, M] Statemet-I is true, Statemet-II is true ; Statemet-II is correct explaatio for Statemet-I. Statemet-I is true, Statemet-II is true ; Statemet-II is NOT a correct explaatio for statemet-i. Statemet-I is true, Statemet-II is false. Statemet-I is false, Statemet-II is true. 7. If the sum of first terms of a A.P. is c, the the sum of squares of these terms is c 6 c c [J 009, M, M] c 8. Let S k, k =,,...,00, deote the sum of the ifiite geometric series whose first term is k k! commo ratio is k 00 00! 00. The the value of 9. Let a,a,a,...,a be real umbers satisfyig a =, 7 a > 0 ad a k = a k a k for k =,,...,. If a a... a 90, the the value of 6 ad the k k S k is [J 0, M] k a a... a is equal to [J 0, M] 0. The miimum value of the sum of real umbers a, a, a,, a 8 ad a 0 with a > 0 is. Let a,a,a,...,a 00 be a arithmetic progressio with a = ad with < < 0, let m =. If S S m p i [J 0,] S a, p 00. For ay iteger p does ot deped o, the a is [J 0, ]. Let a, a, a,... be i harmoic progressio with a = ad a 0 =. The least positive iteger for which a < 0 is [J 0, ( )] i NOD6\\Data\0\Kota\J-Advaced\SMP\Maths\Uit#07\g\0(b)-Sequece-series (xercise).p6. Let k (k ) k S ( ) k. The S ca take value(s) [J-Advaced 0,, ( )] A pack cotais cards umbered from to. Two cosecutive umbered cards are removed from the pack ad the sum of the umbers o the remaiig cards is. If the smaller to the umbers o the removed cards is k, the k 0 = [J- Advaced 0,, ( )] PRVIOUS YARS QUSTIONS ANSWR KY. (a) D, (b) A. (a) A, (b) C, (c) D, (d) A, A,...A H, H,...H XRCIS- [B]. (a) D 6. B 7. = 7 8. C B. D. B. C. A. B 6. C 7. C or. D. A,D.